They were used to protect furniture from marks left by tea cups by providing a coaster for the cup. The tea was often too hot and was poured into a saucer to cool before consumption. Tea cups, at the time, had no handles. In order to drink the tea from the saucer two hands were required which made it necessary to put down the cup. Cup plates provided a spot to put the cup, and were scattered about the parlor during tea parties.
Many were produced, in a variety of designs, from flint glass, the only glass produced by early American glass factories. They are a uniquely American invention for American tea parties, as proper European manners forbade the 'slurping' of cooled tea from tea saucers. European glass factories (Baccarat, Val St. Lambert) produced cup plates for the American market, but the great majority were of American manufacture.
Cup plates were also used to commemorate historical figures, including George Washington and Henry Harrison. These cup plates are called "Historicals." Being enshrined in cup plate history was quite an impressive honor. These are very valuable in modern times. The second type of American cup plates were "Conventionals," or cup plates that simply portrayed the latest in abstract glass design. These were very appealing and "flashy." 
Initial cup plates were made by blowing the glass into the appropriate size and depth (three to four inches diameter, 3/8 to 1/2 inch depth). Later cup plates were produced in pressed glass forms with myriad designs in flint glass, and later in soda glass, before the decline in popularity of tea parties which occurred after the Civil War era. The English Staffordshire potteries also produced large numbers in blue and white transferware, for export.